Contents

# Contents

## Idea

The generalization of the concept of Fourier transform from suitable function to distributions.

## Definition

###### Definition

(Fourier transform of tempered distributions)

For $n \in \mathbb{N}$, let $u \in \mathcal{S}'(\mathbb{R}^n)$ be a tempered distribution. Then its Fourier transform $\hat u \in \mathcal{S}'(\mathbb{R}^n)$ is defined by

$\array{ \mathcal{S}(\mathbb{R}^n) &\overset{\hat u}{\longrightarrow}& \mathbb{C} \\ f &\mapsto& \langle u, \hat f\rangle } \,,$

where on the right $\hat f$ is the ordinary Fourier transform of the function $f$ (an element of the Schwartz space $\mathcal{S}(\mathbb{R}^n)$).

(e.g. Hörmander 90, def. 7.1.9)

## Properties

###### Proposition

The operation of Fourier transform of tempered distributions (def. ) induces a linear isomorphism of the space of temptered distributions with itself:

$\widehat{(-)} \;\colon\; \mathcal{S}'(\mathbb{R}^n) \overset{\simeq}{\longrightarrow} \mathcal{S}'(\mathbb{R}^n)$
###### Proposition

(Fourier transform of compactly supported distributions)

If $u \in \mathcal{D}'(\mathbb{R}^n) \hookrightarrow \mathcal{E}'(\mathbb{R}^n)$ happens to be a compactly supported distribution, regarded as a tempered distribution, then its Fourier transform according to def. is a smooth function

$\hat u \in C^\infty(\mathbb{R}^n)$

given by

$\hat u(k) \;\coloneqq\; u\left( \exp(-i \langle (-), 2 \pi k \rangle) \right) \,,$

where $\langle -,-\rangle$ denotes the canonical inner product on $\mathbb{R}^n$.

This is well-defined also on complex numbers, which makes it an entire holomorphic function (by the Paley-Wiener-Schwartz theorem), called the Fourier-Laplace transform.

(This plays a role for instance in the Paley-Wiener-Schwartz theorem.)

###### Proposition

The Fourier transform (def. ) of the convolution of distributions of a compactly supported distribution $u_1 \in \mathcal{E}'$ with a tempered distribution $u_2 \in \mathcal{S}'$ is the product of distributions of their separate Fourier transforms:

$\widehat {u_1 \star u_2} \;=\; \hat u_1 \hat u_2 \,.$

(Here, by prop. , $\hat u_1$ is just a smooth function, so that the product on the right is just that of a distribution with a function.)